Overview
Roulettes are special types of curves that are made when one curve is rotated around another by a fixed point. They are in Euclidean space, and can be described by calculus equations. Roulettes, cycloids, and other variations of circular movement have applications to pendulums, gears and subatomic particles.
General Definition of a Roulette
A roulette is a type of curve. It is the path of a fixed point in relation to one curve as that curve travels along a straight line or another curve. The simplest form of a roulette is a cycloid. Imagine a circle that looks like a wheel with one spoke from the center to the edge of the wheel, one radius long. The path traced as that circle travels along the line is a cycloid.
The Cycloid Family
If the roulette is formed by a circle travelling outside the circumference of another circle it is called an epicycloid, and if the circle is traveling inside the circumference it is called a hypocycloid. Bernoulli in the late 17th century proposed a challenge, which was solved by him, Newton, and Leibniz. The challenge was to find the fastest path a point will take to travel moving without friction, using gravity.
Playing with Cycloids
Mathematicians play with cycloids using differential calculus equations. The path a cycloid travels can be measured by an equation involving the differential of x divided by the differential of y all squared in relation to the radius of the circle. Many children of all ages have played with a toy called a Spirograph that essentially generates cycloids of various types, using pens of different colors.
Applications
Huygens, living in the late 17th century, used the cycloid to make a pendulum that would be the most efficient type ever devised, in order to improve timekeeping and measurements aboard ship. The motions of gears also follow paths described by the cycloid family. In order for gears to regulate machinery, they need to touch and move freely, along cycloid paths. Engineers calculate the speed of gears; the location, number, and size of teeth (fixed points); and the size of the gears themselves. The path of electrons in electric and magnetic fields also follow cycloid movements.
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